Also on Monday, Surrey police confirmed that they had interviewed Savile under caution in 2007 about sexual assault allegations dating back to the 1970s. No charges were brought after the Crown Prosecution Service advised that there was insufficient evidence to bring a case to court.

That should not come as much of a surprise. Allegations of rape or sexual assault are difficult enough to prove at the best of times, with often little or no corroboration. Savile's alleged victims would have had to satisfy a court that they had a clear recollection of events around 30 years ago. They would also have to explain why they had waited until the former television presenter was in his eighties before coming forward.

Since then, of course, a number of other alleged victims have given interviews to journalists working on television documentaries. If the police had known about these allegations in 2007, that might have strengthened the case against Savile. But the women who came forward after Savile's death last year are unlikely to have known in 2007 that he was under investigation.

Scotland Yard is now facing something of a dilemma. It is too late for Savile to be prosecuted. The police would be accused of wasting scarce resources if they conducted detailed investigations into allegations against a defendant who is no longer able to stand trial.

On the other hand, people want to know whether these allegations can be substantiated. The police would be performing a valuable public service by collecting the available evidence.

In my view, the police need to investigate whether there is evidence against anyone else who might be charged in connection with these allegations. To some extent, that requires detectives to make inquiries into the claims against Savile himself. But that is as far as it should go.

And who might those other defendants be? There is no suggestion that Savile had an accomplice. Some newspapers have pointed the finger at the BBC, both for failing to investigate allegations of abuse by a BBC presenter that may have taken place on BBC premises and for not broadcasting evidence of abuse obtained last year by Newsnight. Perhaps a declaration of interest is needed here: I was on the staff of the BBC for 25 years and still work for the corporation as a freelancer.

The editor of Newsnight has said his team had no new evidence that would have helped the police. He knew the CPS had decided there was insufficient evidence for a prosecution. Whether you think that this was the right editorial decision rather depends on whether you think Newsnight should broadcast a celebrity exposé of an individual who could no longer defend himself. There seem to be no issues of law here.

The next question is whether the BBC was criminally liable for allowing abuse to take place on its premises. Might the BBC have been at fault for permitting vulnerable guests to visit presenters in dressing rooms at Television Centre? Could it be liable as a co-conspirator or even for what used to be called incitement and is now described as intentionally encouraging or assisting the commission of an offence?

Surely not. Imagine the outcry if Auntie had insisted on a chaperone. That would surely have been unsustainable. In general, an omission to act is not in itself a criminal offence. Nor is failing to prevent or report criminal conduct.

There are exceptions, of course: parents are required to care for their children. Organisations that allow public access must ensure people's safety. Employers have duties to their staff. Gross failures in these cases may lead to criminal liability. But I cannot see how the BBC or any of its staff could be prosecuted for failing to stop a presenter from committing offences of this sort on its premises. As far as we know, nobody complained.

Should the BBC have continued to use Savile? Here, the corporation may be on shakier ground. He was not on the staff, of course, but he must have signed countless BBC contracts. The current version of the BBC's standard contract requires presenters to not undermine the BBC's reputation for integrity and decency. The BBC could have sacked him or, at the very least, stopped putting him on the air.

No doubt BBC managers told themselves that it would have been wrong to have dropped a popular presenter because of unsubstantiated allegations about his private life. There might be rather fewer stars on BBC programmes if that approach became widespread. If nothing had been proved, perhaps it would have been unfair, even discriminatory, to have stopped using him?

Perhaps so. But some former BBC bosses must be feeling rather uncomfortable. The BBC is right to cooperate with the police investigation. These days, it is much less willing to condone misbehaviour by people who work for it. Ultimately, though, its job is to make programmes. And it must feel pretty sore about the fact that it missed the chance to make one about Jimmy Savile.